A fully semi-Lagrangian discretization
for the 2D Navier–Stokes equations
in the vorticity–streamfunction formulation
Luca Bonaventura
(1), Roberto Ferretti
(2),
Lorenzo Rocchi
(2)June 14, 2017
(1) MOX – Modelling and Scientific Computing,
Dipartimento di Matematica, Politecnico di Milano Via Bonardi 9, 20133 Milano, Italy
luca.bonaventura@polimi.it
(2) Dipartimento di Matematica e Fisica
Universit`a degli Studi Roma Tre
L.go S. Leonardo Murialdo 1, 00146, Roma, Italy ferretti@mat.uniroma3.it, lor 89@fastwebnet.it
Keywords: Semi-Lagrangian methods, Advection–diffusion equations, Navier– Stokes equations, vorticity–streamfunction formulation.
Abstract
A numerical method for the two-dimensional, incompressible Navier– Stokes equations in vorticity–streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving unconditional stability without the need to solve linear systems beyond that required by the Poisson solver for the reconstruction of the streamfunction. A description of the dis-cretization of Dirichlet boundary conditions for the semi-Lagrangian approach to diffusion terms is also presented. Numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method.
1
Introduction
Over the last 30 years, Semi-Lagrangian (SL) schemes have been ex-tremely successful for advection dominated problems, allowing for example major reductions in the computational cost of operational weather predictions, see e.g. [5], [24]. The potential advantage of SL schemes comes from their unconditional stability with respect to the Courant number, which is in turn achieved by exploiting approxima-tions of the characteristics of the advection equation.
Several extensions of SL schemes to advection–diffusion problems have been proposed in the last two decades. Some of these techniques are based on a splitting of the evolution operator, in which only the advective part is treated in SL form, whereas others treat the diffusion term in a consistently SL form by using multiple characteristics, as it will be shown in Section 3. A review of these proposals is presented, e.g., in [12].
Within the second line of work, which has been traditionally mo-tivated by stochastic arguments, we quote here the seminal paper [8], along with the more recent contributions in [17], [18], [19] and [13]. Due to their stochastic origin, all these works treat diffusion opera-tors in trace form, but extensions to divergence form operaopera-tors has also been proposed in [3], [4]. In these papers, SL methods have proved to be accurate and efficient for linear and nonlinear advection–diffusion problems.
In the present work we are interested in applying the same ap-proach to the incompressible Navier–Stokes (NS) equations
∂u
∂t + (u · ∇) u − ν∆u + ∇p = 0,
∇ · u = 0, (1)
on a bounded domain Ω ⊂ R2, with proper initial and boundary con-ditions. SL techniques of both the forms outlined above have been proposed for this problem, we refer here for example to the papers [16], [29], [31], [30] and the review in [2], in which the application of the SL method is restricted to the advective part, and to [1], [20], which present and analyze a fully SL approach based on a stochas-tic framework. In parstochas-ticular, these latter works present a theorestochas-tical analysis for the time-discrete scheme, although with limited numerical validation.
In the present work, we study the application of a fully SL scheme, much in the same spirit of [1], [20], to the two-dimensional NS equa-tions from a more numerical perspective. We will use the vorticity and streamfunction formulation for simplicity, as a first step towards
the application of the same approach in the context of a projection method for the NS equations in their standard velocity-pressure for-mulation. We will show that the fully SL approach yields an explicit discretization of both advection and diffusion terms with very mild stability restrictions, that can achieve higher order spatial accuracy in a practically and conceptually simple way, while reducing the com-putational cost of the advection–diffusion step. The scheme will be constructed and its consistency analyzed. Moreover, we will provide details on the implementation of boundary conditions, which was not discussed in detail in our previous papers, and perform some classical numerical tests to validate the method. Numerical results show that the method yields good quantitative agreement with reference numer-ical solution of classnumer-ical benchmarks, while allowing the use of time steps several times larger than those of a standard explicit scheme.
The outline of the paper is the following. Section 2 recalls the vorticity–streamfunction formulation of 2D NS equations, along with the related boundary conditions. Section 3 describes the SL advection– diffusion solver, along with its consistency analysis and the imple-mentation of the boundary conditions. A numerical validation of the proposed approach on classical benchmarks is presented in Section 5, while some conclusions and perspectives for future developments are outlined in Section 6.
2
The Navier–Stokes equations in the
vorticity–streamfunction formulation
We briefly recall here the vorticity–streamfunction (VS) formula-tion of the NS equaformula-tions. The basic idea, see [10] for an in-depth discussion, is that any two-dimensional divergence-free vector field u= (u1, u2) can be obtained as u1= ∂ψ ∂x2 , u2 = − ∂ψ ∂x1 , (2)
for a suitable streamfunction ψ. Now, since u is two-dimensional, its vorticity is a scalar given by
ω = ∂u2 ∂x1 −
∂u1
∂x2
, (3)
which, together with (2), gives
On the other hand, by taking the curl of both sides in (1) we get 0 = ∂ω ∂t − ν∆ω + ∂ψ ∂x2 ∂ω ∂x1 − ∂ψ ∂x1 ∂ω ∂x2 = ∂ω ∂t − ν∆ω + u1 ∂ω ∂x1 + u2 ∂ω ∂x2 , (5)
i.e., an advection–diffusion equation for the vorticity. Then, the VS formulation combines (2), (4) and (5) in the form
∂ω
∂t + u · ∇ω − ν∆ω = 0
−∆ψ = ω u= ∇⊥ψ (6) in which the symbol ∇⊥
denotes in compact form the operator defined by (2). Note that the incompressibility condition is ensured by (2) and that the advantage of treating a scalar problem for the vorticity only occurs in two space dimensions.
The precise derivation of boundary conditions for the vorticity– streamfunction formulation is not trivial, see e.g. the discussion in [11], [22]. Here, we will simply note that no slip boundary conditions amount to assigning
ψ = 0, ∇⊥ψ · n = 0, (7) where n denotes the outer normal at the boundary, while more general non homogeneous boundary conditions are assigned by setting non zero values for the normal derivative of the streamfunction.
3
The fully semi-Lagrangian
numeri-cal method
Our aim in this work is to employ the fully SL advection–diffusion method described in [3] in the framework a discretization of the NS equations in VS formulation. The general idea underlying this tech-nique is to approximate the diffusion term by a convex combination of the known values at the locations of a time dependent stencil. Similar ideas, often justified by probabilistic arguments, have been proposed in various independent contexts, see [8], [13], [21], [25].
Given the advection–diffusion equation
ωt+ u · ∇ω − ν∆ω = 0, (x, t) ∈ R2× [0, T ]
we first consider the inviscid case ν = 0. The construction of large time-step schemes for (8) stems from the application of the classi-cal method of characteristics. The system of characteristic curves X(x, t; s) for (8) is defined by the solutions of:
d
dsX(x, t; s) = u(X(x, t; s), s),
X(x, t; t) = x. (9) The solution of (8) is constant along such curves, which means that it satisfies the relationship
ω(x, t) = ω(X(x, t; t − ∆t), t − ∆t). (10) Discretizing the representation formula (10) we obtain the advective SL approximation. More precisely, we denote by ∆t and ∆x respec-tively the time and space discretization steps, with tn = n∆t for
n ∈ [0, T/∆t], and a space grid of points xi. The characteristics
X defined by (9) will be replaced by their numerical approximations X∆. We will also use the shorthand notation
zn+1i = X∆(xi, tn+1; tn) (11)
to denote the foot of the approximate characteristic starting from (xi, tn+1). In this work, we have employed characteristics
approxima-tions based on the standard explicit Euler and Heun schemes. These methods are applied with substepping, as described e.g. in [9], [14], [23]. More precisely, for the approximation of characteristics a time step ∆τ is employed, that is a fraction of the total time step ∆t. This time step is required to comply with a local CFL restriction. As discussed in [23], the value of ∆τ can vary along each approximated characteristic, so that ∆τ << ∆t only in those (usually small) re-gions with really large Courant numbers. During these substeps, the velocity field u is frozen at a constant time level for simplicity. Since employing un entails at most first order accuracy, in the case of the
Heun scheme the extrapolation
un+12 ≈ 3
2u
n−1
2u
n
is employed, as common in the meteorological literature, see e.g. [24]. For inviscid problems with ν = 0, (10) is discretized by replacing the exact upwinding X with X∆, and the value of u at the foot of a
characteristic with an interpolation Ip:
where ωin+1 is the approximation of ω(xi, tn+1), Ωn is the vector of
node values ωn
i, and Ip is an interpolation operator (e.g., a polynomial
interpolation of degree p) which is assumed to satisfy the condition Ip[Ω](xi) = ωi.
As customary in the implementation of semi-Lagrangian schemes, whenever zin+1 falls outside of the computational domain, it is re-defined as the closest boundary point. Even though this implies in principle an error that is first order in time, its impact is minimized by the use of substepping approaches for the approximation of the trajectories, see e.g. the discussion in [23].
When ν 6= 0, the scheme can be modified along the lines proposed in [3], [13], in order to introduce an approximation of the diffusion term. In fact, a first-order consistent discretization of the term ν∆ω is obtained by replacing the interpolation of the numerical solution at zn+1i with an average of interpolated values obtained adding to zin+1 a second displacement of the form
δk = ±δej, (k = 1, . . . , 4)
for all combination of both the sign and the index j = 1, 2, and with δ =√4ν∆t.
Notice that, in an alternative form of the diffusive part of the scheme, the displacements δk could be rather defined as
δk = δ √ 2 ±1 ±1 ,
which corresponds to the definition given in [1, 20]. While this defi-nition is perfectly equivalent in two space dimensions (the stencil of points is simply rotated), in three space dimensions this would require a higher number of interpolations at each node (eight interpolation points instead of six).
The resulting scheme then reads
ωn+1i = 1 4 4 X k=1 Ip[Ωn] zin+1+ δk , (13)
in which zin+1 is defined by (11). In case some of the displacements δk identifies a location out of the computational domain, the
Figure 1: Modification of weights and displacements near the boundary
along the line connecting zin+1 and zin+1± δej. For these points, the
scheme is redefined as ωn+1i = 4 X k=1 αkIp[Ωn] zn+1i + δk , (14) where P4
k=1αk = 1. In order to define the displacements δk and
weights αk in a consistent form, we consider without loss of generality
the simple case depicted in Figure 1. A pure diffusion operator is discretized by the abstract scheme
u(x, tn+1) ≈ 1 4u(x + δe1, tn) + 1 4u(x − δe1, tn) +α+u(x + δ+e2, tn) + α−u(x − δ−e2, tn),
in which we enforce the constraint
δ− ≤ δM, (15)
according to the scheme of Fig. 1. Following [3], consistency is achieved by imposing conditions on the moments of the discrete mass distribution defined by the weights α±
and displacements δ±
. Requir-ing the first and second moment to coincide with the original ones, we obtain the conditions:
α++ α−= 1/2
α+δ+− α−δ− = 0 (16) α+(δ+)2+ α−
(δ−
)2 = 2ν∆t. The first condition is satisfied if
α−= 1 2 − α
Using the relationship in the second equation of (16), we obtain 0 = α+δ+− α−δ−
= α+δ+− (1/2 − α+)δ− which gives in turn
α+= 1 2 δ− δ++ δ−, α − = 1 2 δ+ δ++ δ−. (17)
Using now (17) in the third equation of (16), we obtain
2∆tν = 1 2 δ+δ−2+ δ− δ+2 δ++ δ− = 1 2δ +δ− , that is, δ+= 4∆tν δ− . (18)
To sum up, consistency under the constraint (15) would be obtained by setting δ− = δM, δ+= 4∆tν δM , α−= 1 2 δ+ δ++ δ−, α += 1 2 − α − . (19)
Note that, in general, (16) ensures only consistency of order 1/2 (see the analysis in [3]). Consistency of first order requires the further condition
α+δ+3− α−δ−3 = O(∆t2),
which is false in general for the solution (19). Therefore, we should expect the consistency rate of the diffusive term to drop to ∆t1/2 at
points where the constraint (15) holds.
Denoting by A a numerical approximation of the operator −∆ (e.g., via a five-point discrete Laplacian), by D⊥
an approximation of the operator ∇⊥
(e.g., via centered differences), and by Ψn+1 and Un+1 the vectors of the node values ψn+1
i and un+1i , respectively, the
final form of the scheme is therefore
ωin+1= 4 X k=1 αkIp[Ωn] zin+1+ δk , AΨn+1= Ωn+1, (20) Un+1= D⊥Ψn+1,
with initial conditions given by Ω0 = AΨ0, U0 = D⊥
Ψ0, where Ψ0 denotes the vector of the initial streamfunction values.
Figure 2: Node arrangement for the Thom boundary conditions
For a complete definition of the numerical method, boundary con-ditions for vorticity must be assigned, which do not follow immediately from the boundary conditions for the streamfunction. A wide range of possibilities is discussed in [11]. Here, we will employ one of the simplest formulations, corresponding to the so called Thom bound-ary conditions [26], which are obtained by converting (via Poisson’s equation) boundary conditions for the derivative of ψ into Dirichlet boundary conditions for ω. More specifically, for a linear portion of the boundary on which a tangential (possibly zero) speed U is imposed, the Thom boundary conditions read
ω0 = −
2
∆y2(ψ1− ψ0) +
2U
∆y. (21) A sketch of the reference geometry is given in Fig. 2. Note that the Thom conditions are directly written in terms of the approximate so-lution on the grid, in which the subscript denotes the index of the node with respect to the boundary, so that ω0, ψ0 refer to the
bound-ary and ψ1 to the first internal layer of nodes. Note also that, in (20),
the computation of (21) should be performed at the start of time step n + 1 after the Poisson equation has been solved for Ψn at time step n (or from the initial data if n = 0). The SL solution of the advection– diffusion equation should then be computed only at internal nodes, using the values provided by (21) as boundary data.
4
Consistency analysis of the method
We give now a simplified consistency analysis for the global method, assuming that it is posed on the whole of R2with a uniform orthogonal grid. The analysis of the advection–diffusion step has been carried out elsewhere (see [13, 3]), and shows that the error introduced between
two successive time steps is
ǫ1 ≤ C ∆t2+ ∆xp+1 .
Using a five-point laplacian, the Poisson equation is solved with O(∆x2)
accuracy. On the other hand, since the right-hand side is perturbed by an error ǫ1, the error introduced on the streamfunction is
ǫ2 ≤ C ∆t2+ ∆xp+1+ ∆x2 .
Finally, the error introduced by the operator D⊥
, when implemented by centered differences, is again of order O(∆x2). Since for the order of interpolation we have p ≥ 1, we can drop the term ∆xp+1, and dividing by ∆t we finally obtain the local truncation error
τ (∆x, ∆t) ≤ C ∆t +∆x 2 ∆t ,
which results in a first-order scheme when working at constant Courant number. A comparison with more conventional advection–diffusion solvers (see again [3]) shows that, despite being formally low-order, this strategy provides a good absolute accuracy. In addition, as usual in SL schemes, no stability restrictions on the Courant number are introduced. However, since the numerical stencil consists of points O(√∆t) apart, some care should be taken to avoid under-resolution of the smaller spatial scales. The analysis carried out in [13] shows that, in order to have a smooth numerical domain of dependence at time T , the compatibility condition
ν1/3∆t T2/3∆x2/3 ≪ 1
should be enforced on the discretization steps. In particular, this implies that time steps with Courant numbers quite larger than one are still acceptable, so that the efficiency of the semi-Lagrangian scheme for the advection term is not lost.
5
Numerical results
The method outlined in the previous sections has been imple-mented so far in a rather elementay way, employing structured Carte-sian meshes on a rectangular domain. The boundary conditions de-scribed in section 3 were employed. In one numerical experiments, the mesh employed had constant step in either direction. The Poisson equation was discretized in this case via a standard five-point finite
Nodes u∆t/∆x ν∆t/(2∆x2) L∞
rel. err. L2 rel. err. p 2 50 × 50 2.6 1 1.12 · 10−2 1.68 · 10−2 -100 × -100 2.6 2 5.44 · 10−3 7.54 · 10−3 1.15 200 × 200 2.6 4 2.58 · 10−3 3.57 · 10−3 1.08 Table 1: Convergence test in benchmark with analytic solution
difference scheme, and the operator D⊥
by centered finite differences. For the interpolation operator Ip, the monotonized bicubic interpolant
implemented in the MATLAB native function interp2 was employed. Also piecewise linear interpolation was tested, but this resulted in an overly dissipative method, as well known in the SL literature for the advective terms (see e.g. [24]) and already shown for the diffusive terms in [3]. The corresponding results are not reported in the follow-ing. In the other tests, meshes with non constant steps were employed, in order to achieve higher resolution at the boundaries, where vortic-ity production takes place. In this case, first order finite differences approximations were employed both for the Poisson equation and the D⊥
operator. For both the constant and non constant step cases, the finite difference approximations employed for the Poisson equa-tion yield matrices that coincide with those that would arise from a linear finite element discretization on the same mesh. Furthermore, for the meshes with non constant steps, the the cubic spline option in interp2 was employed.
5.1
Test case with analytic solution
We consider the test case discussed e.g. in [15]. In this test, that was run with ν = 0.02, over the time interval t ∈ [0, 4], the computational domain is given by [0, 2π] × [0, 2π], with periodic boundary condi-tions. The exact solution is given by ω(x, y, t) = sin x sin y exp (−2νt). For this test, Cartesian meshes with constant node spacing were em-ployed, as described above. The first order convergence behaviour is confirmed, but it can be seen that time steps up to four times those of explicit schemes can be used without any loss in accuracy.
5.2
Lid driven cavity
This classical benchmark for numerical approximations of the Navier Stokes equations has been discussed in detail for the vorticity and streamfunction formulation in [11]. Reference solutions obtained with high order spectral methods and other accurate techniques are re-ported, among many others, in [6], [7]. In our assessment, we focus on
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 3: Lid driven cavity benchmark at Re = 100 : a) flow field b) stream-function.
low or moderate values of the Reynolds number. Indeed, it is known that for Re ≈ 8000 the flow becomes turbulent and unpredictable for this test, so that a more sophisticated statistical assessment would have to be carried out. Furthermore, the proposed method is expected to yield significant efficiency gains especially in laminar regimes, in which the stability restrictions due to the viscous term can affect the efficiency of standard explicit methods.
An example of the flow field and streamfunction obtained in the case Re = 100 at steady state (approximately T = 100) is shown in Figure 3. The streamfunction is plotted with the same isoline val-ues as suggested in [7]. The results were obtained using 100 nodes in each direction and a mesh whose 2 nodes closest to the boundary are separated by a spacing smaller than that used in the rest of the domain. The time step employed was such that ν∆t/(2∆x2) ≈ 4, kuk∆t/∆x ≈ 1 for the smallest ∆x values and the solution was com-puted until steady state was achieved up to a tolerance of 10−7
. At a more quantitative level, the maximum horizontal velocity value along the centerline of the cavity was umax = 0.21458, while the maximum
and minimum vertical velocity values along the centerline of the cavity were vmax= 0.17534, vmin = −0.24613, respectively. These values
im-ply a relative error of order 10−3
with respect to the reference steady state solution in [6]. The vorticity value at the center of the cavity was computed as ωcen = 1.13370, which implies a relative error of order
10−2
with respect to the same reference solution.
Results at Re = 1000 are shown instead in 4. Again, the stream-function is plotted with the same isoline values as suggested in [7] and a good general agreement is found between our results and those reported in [6], [7]. At a more quantitative level, we show in Fig-ure 5 the u component of velocity and the vorticity values along
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4: Lid driven cavity benchmark at Re = 1000 : a) flow field b) streamfunction.
the axis of the cavity, as computed by the proposed method (black line) on the same mesh described above with a time step that yields u∆t/∆x ≈ 6, ν∆t/(2∆x2) ≈ 4. The results are compared to those
of reference solutions presented in [6], shown in blue dots and red circles for velocity and vorticity, respectively. It can be seen that also good quantitative agreement is achieved, in spite of the rela-tively coarse mesh, even though the accuracy loss at the boundaries discussed in Section 3 is apparent in the results for the vorticity vari-able. The maximum horizontal velocity value along the axis of the cavity was umax = 0.37487, while the maximum and minimum
verti-cal velocity values along the horizontal centerline of the cavity were vmax = 0.36034, vmin = −0.49989, respectively. The vorticity value
at the center of the cavity was computed as ωcen = 2.02641. These
values imply relative errors of order 10−2
with respect to the reference steady state solution in [6].
6
Conclusions
We have presented a fully semi-Lagrangian, explicit time discretiza-tion method for the Navier Stokes equadiscretiza-tions in the vorticity stream-function formulation, which allows to use time steps much longer than conventional explicit discretizations without the need to solve any lin-ear systems apart from the one resulting from the discretization of the Poisson equation. While only first order in time, the method allows in practice to achieve a sufficient accuracy for many applications, since high order interpolants can be used to reduce numerical dissipation in a very straightforward way.
The effectiveness of the method was demonstrated by numerical re-sults on classical benchmarks of two dimensional incompressible flows.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 10 15 20
Figure 5: Lid driven cavity benchmark at Re = 1000 : a) comparison with u velocity component values of reference solution b) comparison with ω values of reference solutions
In ongoing work, this approach is being extended to the pressure– velocity formulation of the Navier Stokes equations. Furthermore, we are aiming to implement this approach in the framework of the high order adaptive SL-DG discretizations proposed in [27], [28].
Acknowledgements
Part of this research work has been financially supported by the INDAM– GNCS project “Metodi numerici semi-impliciti e semi-Lagrangiani per sistemi iperbolici di leggi di bilancio” and by University Roma Tre.
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